Geometry: making a tangent to two arbitrary circles

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Geometry: making a tangent to two arbitrary circles

Postby platyhiker » Sat Mar 05, 2011 12:40 am UTC

This is a problem from my high school geometry class, back in the early eighties. I wasn't able to solve it then, and it still bugs me.

Problem: Using a compass and an (unmarked) straight edge, draw a line that is tangential to two arbitrary circles.

If you simplify the problem by making the two circles have the same diameters, the problem is easy:
1) draw a line through the two centers of the circles
2) at each circle center, draw a line perpendicular to the line in step 1
3) draw a line connecting where the lines in step 2 intersect the circles

Once I make the circles have different diameters, I can't seem to figure out how to solve the problem.

One interesting thing is that it's not terribly hard to *experimentally* come up with a line that is tangential to two arbitrary circles - just keep adjusting the straight edge bit by bit until you've got it touching each circle at exactly one point. But, that wasn't the goal of the problem.

Thank you for your help!
Last edited by platyhiker on Tue Mar 08, 2011 1:52 am UTC, edited 1 time in total.
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Re: Geometry: making a tangent to two arbitrary circles

Postby jaap » Sat Mar 05, 2011 7:11 am UTC

I don't know the full solution, but have a suggestion for the first few steps.

Do your steps 1-3. As you said, the resulting line is not the tangent unless the circles are the same size.
Find the intersection of this line with the line that goes through the two circle centres.
The tangent line you are looking for also has to go through this intersection point. (proof using similar triangles omitted)
So now you have reduced the problem from "a tangent to 2 circles" to "a tangent to 1 circle, also going through a given point".
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Re: Geometry: making a tangent to two arbitrary circles

Postby greengiant » Sat Mar 05, 2011 2:41 pm UTC

As was pointed out to me recently when I said something similar, two arbitrary circles don't always have a common tangent. You need to make sure one circle is not fully inside the other.

Anyway, here's a way to do it http://web.mat.bham.ac.uk/C.J.Sangwin/T ... mpass.html Took a minute for the java applet to load for me.

P.S. If someone had asked me this, I'd have just lined the straight-edge up with the two circles and drawn a line. Not sure why that constitutes cheating - not that different from lining the straight-edge up with two points and connecting them.
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Re: Geometry: making a tangent to two arbitrary circles

Postby ConMan » Mon Mar 07, 2011 1:08 am UTC

greengiant wrote:As was pointed out to me recently when I said something similar, two arbitrary circles don't always have a common tangent. You need to make sure one circle is not fully inside the other.

Anyway, here's a way to do it http://web.mat.bham.ac.uk/C.J.Sangwin/T ... mpass.html Took a minute for the java applet to load for me.

P.S. If someone had asked me this, I'd have just lined the straight-edge up with the two circles and drawn a line. Not sure why that constitutes cheating - not that different from lining the straight-edge up with two points and connecting them.

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More specifically, the idea of compass and straightedge constructions is that your straightedge can only draw lines between two previously identified points, and points can only be identified by intersections of lines and/or circles. So you don't know where on the circles you're meant to line your straightedge up with to draw the tangents until you've done something to find the exact points.
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Re: Geometry: making a tangent to two arbitrary circles

Postby greengiant » Mon Mar 07, 2011 12:30 pm UTC

Yeah, that's fair enough. I get that the compass and straightedge game is played to certain official rules which don't allow this sort of thing. It still feels very restrictive though.

I'm probably being silly, but I'll try to explain. I guess I generally take the view that if you can find some way to make better use of your compass and straightedge, more power to you. I'd agree that any form of guesswork isn't on, but I don't see why lining your edge up with two circles is inherently inaccurate. I don't like the idea of saying 'you're given a compass and straightedge' then saying 'stop, you weren't meant to use it like that' - feels like reprimanding someone for a clever idea.

Maybe my problem is thinking in terms of an idealised compass and an idealised straightedge, when I should think in terms of an arc-producing machine and a machine that joins two points with a straight line.
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Re: Geometry: making a tangent to two arbitrary circles

Postby Talith » Mon Mar 07, 2011 6:11 pm UTC

You should think about the problem as using a set of axioms which you can not work out of. This is important because there are important fields of numbers and structures that are made from these kinds of axioms (e.g the constructable numbers). So although you might say "well hey, why can't I just do this?" there is normally a pretty good reason why you can't.
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Re: Geometry: making a tangent to two arbitrary circles

Postby greengiant » Mon Mar 07, 2011 7:06 pm UTC

Yeah, like I said, I'm probably being silly. I understand the concept of axiomatic systems, I've actually done a fair amount of set theory/logic/model theory.

Maybe it'd be more clear to say I find it mentally jarring not to be able to use my straightedge that way. I understand it's not in the rules.
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Re: Geometry: making a tangent to two arbitrary circles

Postby platyhiker » Tue Mar 08, 2011 1:51 am UTC

jaap - It turns out your suggestion is correct. Interestingly, I think did try that approach when I originally got the problem, but it didn't seem to work then - I guess I got bitten by lack of precision of my tools and drawing skills. Since your post, I tested the idea algebraically (with a specific set of circles) and found that the tangent did go through the intersection point.

Another interesting thing is that I do NOT remember enough geometry to be able to solve the problem of "draw a tangent to 1 circle, going through a given point", and I'm not at all sure I ever did.

greengiant - thank you for URL for a full solution. I don't think I could ever come up with that on my own. For anybody who wants to solution without following the link, here's my write up of how to get one of the tangents:

0) Draw your circles and mark their centers
1) Determine the difference between the two circles radii. (Call it D)
2) Draw a circle of radius D around the center of the larger circle
3) Draw a line between two circle centers
4) Determine the midpoint of the line (Call it O)
5) At O, draw a circle that goes through the center of each starting circle
6) Mark points that step 2 circle intersects with step 5 circle
7) Draw line from one of those points to center of smaller circle
8) Draw line from same point that is perpendicular to step 7 line
9) Mark where step 8 line intersects larger circle
10) At step 9 point, draw line perpendicular to step 8 line.

(The Java applet at the link covers how to get all 4 of the possible tangents.)

As for the discussion on what is allowable with straight edge and compass problems, I look at it as there is certain beauty/elegance to defining a system of rules and determining what is and is not possible within that system. For the straight edge and compass set of problems, you can do lots of neat things - duplicate angles, lengths, make parallel lines, make perpendicular lines, bisect angles and lines, and yet, there are things that are impossible - trisect an arbitrary angle, double a cube, square a circle. In college, I did a second semester of abstract algebra and near the end of semester we were able to prove the impossibility of those constructions. I don't remember any details of the proof, but I remember finding it very, very elegant and very interesting.

Thank you again for helping with a problem that has bothered me for years!
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Re: Geometry: making a tangent to two arbitrary circles

Postby antonfire » Tue Mar 08, 2011 7:41 pm UTC

One justification for forbidding lining up the straightedge with two circles but allowing it with two points is that, with two points, you can first line up the straightedge with one, then pivot it around the point until it hits the other. That is, it's essentially two one-dimensional problems.

You can do something similar with circles, first line it up to be tangent to your circle, then pivot around the center of that circle until it is tangent to the other. The difference is that you are pivoting around a point which is not on the straightedge, a less "physical" operation.
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Re: Geometry: making a tangent to two arbitrary circles

Postby quintopia » Tue Mar 08, 2011 11:14 pm UTC

Unless you have a compass with a right angle bend in it! (For instance, a square with a sliding pivot ring)

Any guesses what the constructible numbers are if you add such a device?
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Re: Geometry: making a tangent to two arbitrary circles

Postby jestingrabbit » Tue Mar 08, 2011 11:19 pm UTC

quintopia wrote:Any guesses what the constructible numbers are if you add such a device?


I'd expect them to be the same as before. We can construct a tangent to two circles, and that seems to be the only "extra" that this tool offers.
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Re: Geometry: making a tangent to two arbitrary circles

Postby quintopia » Wed Mar 09, 2011 12:11 am UTC

and what if you could use a trammel? still the same?
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Re: Geometry: making a tangent to two arbitrary circles

Postby jestingrabbit » Wed Mar 09, 2011 12:52 am UTC

Off the top of my head... no idea. They only help you draw ellipses? If you require that the length of the axes are constructible, which seems fair, you only get more when you intersect ellipses, which could be reduced to the case of an ellipse and a circle by dilating a coordinate. The question is do the coordinates of an intersection of that kind represent roots of general quartics, some class of quartics, or are they something of lower degree, disguised as something trickier?

At first glance of some algebra I just scribbled out, the coordinates seem to solve quartics, but from theory its clear that they aren't general quartics (they can't be imaginary). But my gut is saying that they do allow more complicated numbers, so, plug

Code: Select all
http://www.wolframalpha.com/input/?i=solve%282*x^2+%2B+y^2+%3D+1%2C+%28x-1%29^2+%2B+%28y-1%29^2+%3D+1%29


into alpha. Note that the nontrivial real solution has cube roots.

Answer: you get more numbers when you allow a trammel.
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